Why is x^0 = 1?

When b is a positive integer, is defined to be the product
of a multiplied by itself b times.

The question is, what is the most natural way to extend this definition
to the case when b=0? Here are several ways to see that the
definition is the only reasonable one:

Exponentiation satisfies the laws of exponents: . If we want this law to still be satisfied when we extend
to the case b=0, we need to have
, and therefore we need to have
.

If is b copies of the number a, all multiplied
together, then should be the "empty product" with no
factors multiplied together. In mathematics, the empty product is
defined to be 1, because multiplying by nothing at all is the same
as multiplying by 1.

Notice that can be thought of as "start with the number 1,
then multiply by a, b times." For instance,
and . Therefore, should be just 1, not multiplied
by anything else at all.

When a is a positive integer, yet another reason for
defining is that is the number of ways of
writing (in order) b numbers, each from 1 to a.
For instance, because there are nine different pairs of numbers
each of which is in the range from 1 to 3 (they are (1,1), (1,2), (1,3),
(2,1), (2,2), (2,3), (3,1), (3,2), and (3,3)).

Therefore, should be the number of ways of writing no numbers,
each of which is from 1 to a. There is exactly one way of doing this,
namely, don't write any numbers at all!

(This reason is more compelling if you make it more mathematically precise,
using the fact that is the number of functions from a
b-element set B to an a-element set A, and when b=0 the set B
is the empty set, and there is exactly one function from the empty set into
A, namely, the empty function).

The above reasons all illustrate why defining to be 1 is the
only reasonable definition.

There's one other point worth mentioning: some of the reasons above
are less compelling when a=0. For instance, in the first reason, we
need to have , and if a is non-zero we can divide by
to deduce that . However, if a=0 we no longer get
a reason for to be 1.

Some of the reasons are still compelling, and, especially if we are in
a context where only integer exponents are being considered, we still
normally define to be 1.

However, if we define a two-variable function ,
then this function does not have a well-defined limit as
(x,y) -> (0,0). We can define if we like,
but the limit still won't exist. In other words,
if A and B each approach zero, there's no guarantee as to
what (if anything) approaches. It need not approach
our definition of .

That's why, in calculus,
is often called an indeterminate form. If one is working
in situations where the exponent can continuously vary, it is usually
better to leave undefined to avoid making mistakes. However,
if one is working in situations in which the exponent is always
integral, is usually defined to be 1.

These complications are only for . When a is nonzero,
is always defined to be 1, for the reasons given above.